Question

The number of elements in the set $\{n\in \{1,2,3,\dots \dots ,100\}$ $(11{)}^n>(10{)}^n+(9{)}^n\}$ is _______.

Answer 1

Answer: 96

$11^n>10^n+9^n$
$\Rightarrow 11^n-9^n>10^n$
$\Rightarrow (10+1{)}^n-(10-1{)}^n>10^n$
$\Rightarrow \left\{{}^n{C}_1\cdot 10^{n-1}+{}^n{C}_{3}10^{n-0}+{}^n{C}_{5}10^{n-5}+\dots \dots \right\}>10^n$
$\Rightarrow 2n\cdot 10^{n-1}+2\left\{{}^n{C}_{3}10^{n-3}+{}^n{C}_{5}10^{n-5}+\dots \dots .\right\}>10^n$ .... (1)
For $n=5$
$10^5+2\left\{{}^5{C}_{3}10^2+{}^5{C}_5\right\}>10^5$ (True)
For $n=6,7,8,\dots \dots .100$
$2n10^{n-1}>10^n$
$\Rightarrow 2n10^{n-1}+2\{{}^n{C}_{3}10^{n-3}+{}^n{C}_{5}10^{n-5}+\dots \dots ..3>10^{\circ }$
$\Rightarrow 11^n-9^n>10^n$ For $n=5,6,7,\dots \dots .100$
For $n=4,$ Inequality (1) is not satisfied
$\Rightarrow$ Inequality does not hold good for
$N=1,2,3,4$
So, required number of elements $=96$